Comments on the holographic description of Narain theories

TL;DR

This paper explores the holographic duality of Narain CFTs, proposing bounds on their spectral gap and analyzing the density of states, with implications for quantum stabilizer codes and the consistency of the holographic description.

Contribution

It introduces a hypothesis relating Narain theories to $U(1)$ gravity and derives bounds on spectral gaps and state density variance, supported by analyses of lattice CFTs and quantum codes.

Findings

Bound on spectral gap: $rac{c}{2 extpi e}$

Density of states variance is exponentially small in $c$

New bounds on quantum stabilizer codes compatible with existing literature

Abstract

We discuss the holographic description of Narain $U(1)^c\times U(1)^c$ conformal field theories, and their potential similarity to conventional weakly coupled gravity in the bulk, in the sense that the effective IR bulk description includes "$U(1)$ gravity" amended with additional light degrees of freedom. Starting from this picture, we formulate the hypothesis that in the large central charge limit the density of states of any Narain theory is bounded by below by the density of states of $U(1)$ gravity. This immediately implies that the maximal value of the spectral gap for primary fields is $\Delta_1=c/(2\pi e)$. To test the self-consistency of this proposal, we study its implications using chiral lattice CFTs and CFTs based on quantum stabilizer codes. First we notice that the conjecture yields a new bound on quantum stabilizer codes, which is compatible with previously known bounds in the literature. We proceed to discuss the variance of the density of states, which for consistency must be vanishingly small in the large-$c$ limit. We consider ensembles of code and chiral theories and show that in both cases the density variance is exponentially small in the central charge.